MHB Evaluate the double sum of a product

[lfdahl]

Evaluate the following double sum of a product:

$$\sum_{j=1}^{\infty}\sum_{n=1}^{\infty}\left(n\prod_{i=0}^{n}\frac{1}{j+i}\right)$$

 

[lfdahl]

Hint:

Spoiler

The answer is: $e-1.$

 

[lfdahl]

Suggested solution:

Spoiler

Let
\[\alpha_j(n) = \frac{1}{j(j+1)(j+2)...(j+n)}\]
and let
\[\beta_j(n) = \frac{1}{j(j+1)(j+2)...(j+n-1)}\]

Consider the difference:

\[\beta_j(n)-\beta_{j+1}(n) = \frac{1}{j(j+1)(j+2)...(j+n-1)}-\frac{1}{(j+1)(j+2)(j+3)...(j+n)} \\\\ = \frac{j+n-j}{j(j+1)(j+2)...(j+n)} \\\\ = n \alpha_j(n)\]

Now
\[\sum_{j=1}^{\infty} \alpha_j(n) = \frac{1}{n}\sum_{j=1}^{\infty}\left ( \beta _j(n)-\beta _{j+1}(n) \right )\]

is a telescoping sum, and we get (the limit of $\beta$ is zero):

\[\sum_{j=1}^{\infty} \alpha_j(n) = \frac{1}{n}\left ( \beta _1(n)-\lim_{j \to \infty}\beta _j(n) \right ) =\frac{\beta _1(n)}{n}= \frac{1}{n \cdot n!}\]

Finally, we´re able to evaluate the given double sum above:

\[\sum_{j=1}^{\infty}\sum_{n=1}^{\infty}\left (n\prod_{i=0}^{n}\frac{1}{j+i} \right ) = \sum_{n=1}^{\infty}n \sum_{j=1}^{\infty} \alpha _j(n) = \sum_{n=1}^{\infty}\frac{1}{n!} = e-1.\]